Theorem r0cld 23725

Description: The analogue of the T₁ axiom (singletons are closed) for an R₀ space. In an R₀ space the set of all points topologically indistinguishable from 𝐴 is closed. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypothesis

Ref	Expression
kqval.2	⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})

Assertion

Ref	Expression
r0cld	⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Distinct variable groups:   𝑥,𝑜,𝑦,𝑧,𝐴   𝑜,𝐽,𝑥,𝑦,𝑧   𝑜,𝐹,𝑧   𝑜,𝑋,𝑥,𝑦,𝑧

Allowed substitution hints:   𝐹(𝑥,𝑦)

Proof of Theorem r0cld

Step	Hyp	Ref	Expression
1		kqval.2	. . . . . 6 ⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦})
2	1	kqffn 23712	. . . . 5 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋)
3	2	3ad2ant1 1140	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 Fn 𝑋)
4		fncnvima2 7006	. . . 4 ⊢ (𝐹 Fn 𝑋 → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
5	3, 4	syl 17	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}})
6		fvex 6844	. . . . . 6 ⊢ (𝐹‘𝑧) ∈ V
7	6	elsn 4573	. . . . 5 ⊢ ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ (𝐹‘𝑧) = (𝐹‘𝐴))
8		simpl1 1199	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐽 ∈ (TopOn‘𝑋))
9		simpr 486	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝑧 ∈ 𝑋)
10		simpl3 1201	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → 𝐴 ∈ 𝑋)
11	1	kqfeq 23711	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦)))
12		eleq2w 2825	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝑧 ∈ 𝑦 ↔ 𝑧 ∈ 𝑜))
13		eleq2w 2825	. . . . . . . . 9 ⊢ (𝑦 = 𝑜 → (𝐴 ∈ 𝑦 ↔ 𝐴 ∈ 𝑜))
14	12, 13	bibi12d 347	. . . . . . . 8 ⊢ (𝑦 = 𝑜 → ((𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
15	14	cbvralvw 3219	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐽 (𝑧 ∈ 𝑦 ↔ 𝐴 ∈ 𝑦) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜))
16	11, 15	bitrdi 289	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑧 ∈ 𝑋 ∧ 𝐴 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
17	8, 9, 10, 16	syl3anc 1380	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = (𝐹‘𝐴) ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
18	7, 17	bitrid 285	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ {(𝐹‘𝐴)} ↔ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)))
19	18	rabbidva 3399	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ (𝐹‘𝑧) ∈ {(𝐹‘𝐴)}} = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
20	5, 19	eqtrd 2776	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) = {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)})
21	1	kqid 23715	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
22	21	3ad2ant1 1140	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽)))
23		simp2 1144	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ Fre)
24		simp3 1145	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → 𝐴 ∈ 𝑋)
25		fnfvelrn 7025	. . . . . 6 ⊢ ((𝐹 Fn 𝑋 ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
26	3, 24, 25	syl2anc 591	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ran 𝐹)
27	1	kqtopon 23714	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
28	27	3ad2ant1 1140	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹))
29		toponuni 22901	. . . . . 6 ⊢ ((KQ‘𝐽) ∈ (TopOn‘ran 𝐹) → ran 𝐹 = ∪ (KQ‘𝐽))
30	28, 29	syl 17	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → ran 𝐹 = ∪ (KQ‘𝐽))
31	26, 30	eleqtrd 2843	. . . 4 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽))
32		eqid 2741	. . . . 5 ⊢ ∪ (KQ‘𝐽) = ∪ (KQ‘𝐽)
33	32	t1sncld 23313	. . . 4 ⊢ (((KQ‘𝐽) ∈ Fre ∧ (𝐹‘𝐴) ∈ ∪ (KQ‘𝐽)) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
34	23, 31, 33	syl2anc 591	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽)))
35		cnclima 23255	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ {(𝐹‘𝐴)} ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
36	22, 34, 35	syl2anc 591	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → (◡𝐹 “ {(𝐹‘𝐴)}) ∈ (Clsd‘𝐽))
37	20, 36	eqeltrrd 2842	1 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (KQ‘𝐽) ∈ Fre ∧ 𝐴 ∈ 𝑋) → {𝑧 ∈ 𝑋 ∣ ∀𝑜 ∈ 𝐽 (𝑧 ∈ 𝑜 ↔ 𝐴 ∈ 𝑜)} ∈ (Clsd‘𝐽))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 397   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121  ∀wral 3055  {crab 3393  {csn 4558  ∪ cuni 4841   ↦ cmpt 5156  ^◡ccnv 5620  ran crn 5622   “ cima 5624   Fn wfn 6484  ‘cfv 6489  (class class class)co 7360  TopOnctopon 22897  Clsdccld 23003   Cn ccn 23211  Frect1 23294  KQckq 23680

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-ov 7363  df-oprab 7364  df-mpo 7365  df-map 8769  df-qtop 17466  df-top 22881  df-topon 22898  df-cld 23006  df-cn 23214  df-t1 23301  df-kq 23681

This theorem is referenced by:  nrmr0reg  23736